Recognising conjugacy classes of Dehn twists on $\mathbb D_3$

TL;DR

This paper studies the action of Dehn twists on a thrice-punctured disk, interpreting their dynamics via homology and providing an explicit orbit description and an untwisting algorithm for conjugacy classification.

Contribution

It offers a detailed analysis of Dehn twist conjugacy classes on $ ext{D}_3$, including an explicit orbit description and an untwisting algorithm for minimal factorization.

Findings

Explicit orbit descriptions of Dehn twist actions

An untwisting algorithm for conjugacy problem

Solution to the conjugacy problem for Dehn twists

Abstract

We analyse the action of the basic Dehn twists on the essential curves, $\gamma$, in a disc with 3 marked points, $\mathbb D_3$. In particular, we interpret the induced dynamics on the Dynnikov plane in terms of the standard dynamics in homology $H_1({\rm T})=\mathbb Z^2$ of the branched covering torus with a hole, ${\rm T}\to \mathbb D_3$. Our explicit description of orbits of the action of the pure mapping class group ${\rm PMod}(\mathbb D_3)$ can be viewed as a solution of the conjugacy problem for the Dehn twists $t_{\gamma}$. We also present an ``untwisting algorithm'' for factorization of this problem into a minimal number of steps.